Explicit Lie bracket of closed geodesics on a hyperbolic surface with applications

TL;DR

This paper introduces a geometric method to define the Goldman bracket for closed geodesics on hyperbolic surfaces, enabling algebraic characterization of simple closed geodesics without self-intersections.

Contribution

It develops a non-Euclidean geometric toolbox to construct the Goldman bracket using closed geodesics, providing a new algebraic perspective.

Findings

Constructive geometric definition of the Goldman bracket

Algebraic characterization of simple closed geodesics

Application of non-Euclidean geometry methods

Abstract

In this note we develop a tool box of non-Euclidean plane geometry methods that yield a constructive way to define in terms of closed geodesics the Goldman bracket on deformation classes of closed, directed curves. We use this construction to algebraically characterize closed geodesics without self-intersection on hyperbolic surfaces.